Recall from Proposition REF  that in this case the \(H\) -action on \(X_\mathcal {C}\)  is decomposable with trivial morphism, i.e.  there exists a standard trivialization \(X \simeq G/Q \times Q/R\)  for which the associated cocycle restricted to \(\mathcal {C}\)  is the trivial morphism. Therefore in this case we have \(\mu ^{\ast k} \ast \delta _{(\theta ,f)}=\int \delta _{g \theta } d\mu ^{\ast k}(g) \otimes \delta _f\) , in other words, the equidistribution problem is only the one in \(\mathcal {C} \subseteq G/Q\) . It is well-known that by spectral gap property we have the convergence \(\int \delta _{g \theta } d\mu ^{\ast k}(g) \rightarrow \overline{\nu }_F\)  moreover with exponential speed estimates with respect to a class of Hölder functions. We omit the statement to avoid repetition; see [1]}.
